Hilbert squares of genus 16 K3 surfaces

TL;DR

This paper explores the geometry of genus 16 K3 surfaces, establishing isomorphisms between their Hilbert squares and moduli spaces of stable sheaves, and constructs a 3-form potentially linking to Debarre-Voisin fourfolds.

Contribution

It proves the isomorphism between the Hilbert square of a genus 16 K3 surface and a moduli space of stable sheaves, and constructs a 3-form suggesting a geometric link to Debarre-Voisin fourfolds.

Findings

S^{[2]} is isomorphic to M_{S'}(2,h',7)

Explicit embeddings of vector bundles into hyper-Kähler fourfolds

Construction of a 3-form potentially linking to Debarre-Voisin fourfolds

Abstract

We consider the geometry of a general polarized K3 surface $(S,h)$ of genus 16 and its Fourier-Mukai partner $(S',h')$. We prove that $S^{[2]}$ is isomorphic to the moduli space $M_{S'}(2,h',7)$ of stable sheaves with Mukai vector $(2,h',7)$ and describe the embeddings of the projectivization of the stable vector bundle of Mukai vector $(2,-h',8)$ over $S'$ into these two isomorphic hyper-K\"ahler fourfolds. Following the work of Fr\'ed\'Eric Han in arXiv:2501.16013, we explicitly construct an interesting 3-form $t_1\in \wedge^3 V_{10}^*$ which potentially gives an isomorphism between $S^{[2]}$ and the Debarre-Voisin fourfold in $G(6,V_{10})$ associated to $t_1\in \wedge^3 V_{10}^*$. This would provide a geometric explanation of the existence of such an isomorphism, which was proved in arXiv:2102.11622 by a completely different argument.